Lenz's Law

Problems from IIT JEE

Problem (IIT JEE 1997): An infinitesimally small bar magnet of dipole moment $\vec{M}$ is pointing and moving with the speed $v$ in the positive $x$ direction. A small closed circular conducting loop of radius $a$ and negligible self-inductance lies in the $y\text{-}z$ plane with its centre at $x=0$, and its axis coinciding with the $x$-axis. Find the force opposing the motion of the magnet, if the resistance of the loop is $R$. Assume that the distance $x$ of the magnet from the centre of the loop is much greater than $a$.

Solution: Lenz's Law The magnetic field due to an infinitesimal bar magnet for end-on position is given by, \begin{alignat}{2} & \vec{B}=\frac{\mu_0}{4\pi}\frac{2M}{x^3}\,\hat\imath, \nonumber \end{alignat} where $M$ is magnetic moment and $x$ is the distance from the magnet. Since distance $x$ is much greater than radius~$a$ of the loop, magnetic field can be taken as uniform throughout the loop. For the loop, flux $\phi$, induced emf $e$, induced current $i$ and the magnetic moment $M^\prime$ are given by, \begin{alignat}{2} &\phi=\vec{B}\cdot\vec{S}=\frac{\mu_0}{4\pi}\frac{2M}{x^3}(\pi a^2)=\frac{\mu_0a^2M}{2x^3}, \nonumber\\ &e=-\frac{\mathrm{d}\phi}{\mathrm{d}t}=\frac{3\mu_0 a^2M}{2x^4}\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{3\mu_0 a^2Mv}{2x^4}, \nonumber\\ &i=\frac{e}{R}=\frac{3\mu_0 a^2Mv}{2Rx^4}, \nonumber\\ &M^\prime=iS=\frac{3\pi\mu_0 a^4Mv}{2Rx^4}. \nonumber \end{alignat} According to Lenz's law, the direction of $M^\prime$ is such that it opposes approaching magnet (see figure). The potential energy of the loop having magnetic moment $M^\prime$ and placed in magnetic field $\vec{B}$ is, \begin{alignat}{2} U&=-\vec{M^\prime}\cdot \vec{B}=M^\prime B=\frac{3\pi\mu_0 a^4Mv}{2Rx^4}\,\frac{\mu_0}{4\pi}\frac{2M}{x^3}=\frac{3}{4}\frac{\mu_0^2a^4M^2v}{Rx^7}, \end{alignat} and the force acting on the loop is, \begin{alignat}{2} F&=-\frac{\mathrm{d}U}{\mathrm{d}x}=\frac{21}{4}\frac{\mu_0^2a^4M^2v}{Rx^8}. \nonumber \end{alignat} The positive sign confirms the repulsive nature of force.